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Chapter 12 Notes
Algebra 2
Name ____________________________________
Period _____
12.1 The Fundamental Counting Principle and Permutations
Ex. 1 How many ways can you make a salad given: 2 types of lettuce: romaine and
iceberg and four types of dressing: ranch, 1000, blue cheese, and Italian.
Tree Diagram:

Fundamental Counting Principle: 2 or more events =
o 2: if an event occurs m ways and another event occurs n ways, then the
number of ways that both can occur is m n .
o 3 or more: the product of all the number of ways is m n o ... z .
Ex. 2 At a restaurant, you have a choice of 8 different entrees, 2 different salads, 12
different drinks, and 6 different desserts. How many different dinners, given one of each,
can you choose?
Ex. 3 Consider a standard license plate with 7 items. If the first 4 items were numbers
and the last 3 items were letters:
A) How many different plates are possible if both numbers and letters can repeat?
B) How many different plates are possible if both numbers and letters can’t repeat?
Ex. 4 How many different 7 digit phone numbers are possible if the first digit cannot be
0 or 1?

Permutation: an ordering of n objects  n!  n  n 1  n  2  n  3 ... 3 2 1
o ie: number of ways to arrange the numbers 1, 2, 3 = 3! 3 2 1  6 ways

Definition: n !  1 2 3 4 ... n and 0!  1 .
Ex. 5 You have homework assignments from 5 different classes to complete this
weekend.
A) In how many different ways can you complete the assignments?
B) In how many different ways can you choose 2 of the assignments to complete first
and last?

For part B: the number of permutations of 5 objects taken 2, is
5!
denoted by 5 P2 and given by
.
 5  2 !
* Permutations of n objects taken r at a time: (________________ matters!)
n Pr


!

!
Ex. 6 There are 12 books on the summer reading list. You want to read some or all of
them. In how many orders can you read
A) 4 of the books?
B) All 12 of the books?

During a permutation, the objects are not always distinct. Some objects may
repeat.
o Ie: mom:
mom omm mmo
 m o m, o m m, m m o
mom omm mmo
* Permutations with Repetition: where one object is repeated q1 times, another is
repeated q2 times and so on.
n!
q1 ! q2 ! ... qk !
Ex. 7 Find the number of distinguishable permutations of the letters in
A) Summer
B) Waterfall
C) Mississippi
D) Word
12.2 Combinations and the Binomial Theorem

Combination: is a selection of r objects from a group of n objects where order is
_________ important.
* Combination: (order does _______ matter) ie. a,b,c and b,c,a are the same
n Cr


!
 !
!
Ex. 1 Use a standard deck of 52 cards.
A) If order is not important, how many different 7-card hands are possible?
B) How many of these 7-card hands have all 7 cards of the same suit?
C) How many 5 card hands contain exactly 3 kings?

Number of ways:
o event a ___________ event b occurs

You _______________________
o event a ________ even b occurs

You ___________
Ex. 2 You are taking a vacation. You can visit as many as 5 different cities and 7
different attractions.
A) Suppose you want to visit exactly 3 different cities and 4 different attractions.
How many different trips are possible?
B) Suppose you want to visit at least 8 locations (cities or attractions). How many
different types of trips are possible?
Ex. 3 A restaurant offers 6 salad toppings. On a deluxe salad, you can have up to 4
toppings. How many different combinations of toppings can you have?

4
Pascal’s Triangle: an arrangement of n C r :
0
C0
1
C0
1
C1
2
C0
2
C1
2
3
C0
3
C1
C0
4
C1
4
3
C2
C2
C2
4
3
C3
C3
4
C4
12.2 (Part 2)

Binomial Theorem: the expansion of  a  b  :
n
 a  b  n C 0 an b0  n C 1 an 1 b1  n C 2 an 2 b2  ...  n C n a0 bn
n
n
  n C r a n  r br
r 0
Ex. 4 Expand  a  3
5
Ex. 5 Expand  a  2b3 
4
Ex. 6 Expand  x  5
4
Ex. 7 Expand  2x  y 2 
3
Ex. 8 Find the coefficient of x7 in  2  3x  .
10
Ex. 9 Find the coefficient of x4 in  2 x  3 .
12
12.3 An Introduction to Probability

Probability: a number between ___ and ___ that indicates the likelihood the
event will occur (answer can be written is fraction, decimal, or percent).
o probability = what you ___________ over the _____________ thing.


P = ___: event can’t occur
P = ___: event always occurs
o Theoretical Probability: when all outcomes are _______________ likely.
o
P a 
number of outcomes in A
total number of outcomes
o Experimental Probability: by an experiment, survey, or looking at the
history of the event.
o Geometric Probability: calculating a ratio of 2 lengths, areas, or
volumes.
Ex. 1 A spinner has 8 equal-size sectors numbered from 1 to 8. Find the probability of:
A) spinning a 6
B) spinning a number greater than 6.
Ex. 2 There are 9 students on the math team. You draw their names one by one to
determine the order in which they answer questions at a math meet. What is the
probability that 3 of the 5 seniors on the team will be chosen last, in any order?
Ex. 3 Find the probability that a randomly chosen student from this year’s 9th grade
class is enrolled in:
A) Consumer Math
B) Algebra 1 or Introduction to Algebra
Ex. 4 Find the probability that a dart would hit the shaded region.
Ex. 5 Five cards are drawn from a 52-card deck. What is the probability the first two
cards are red?
Ex. 6 Find the probability of drawing the given card from a standard 52-card deck:
a) the queen of hearts
b) a non-face card
12.4 Probability of Compound Events

Compound Event: the union or intersection of 2 events.
o Union: all outcomes of ____________ A and B.
o Intersection: only _______________ outcomes of A and B.

Mutually Exclusive Events: when there is ______ outcomes in the intersection
of A and B.

Probability of Compound Events:
o
 or 
 and 
P A
B   P  A  P  B   P  A
B
 
  

o
P  A  B   P  A  P  B   no intersection (mutually exclusive)
P  A  B  0
Ex. 1 One six-sided die is rolled. What is the probability of rolling a multiple of 2 or a
5.
Ex. 2 A card is randomly selected from a standard deck of 52 cards. What is the
probability that the card is a heart or a number less than 3 (including the ace)?
Ex. 3 In a poll of high school juniors, 6 out of 15 took a French class and 11 out of 15
took a math class. Fourteen out of 15 students took French or math. What is the
probability that a student took both French and math?

Complement of Event A: (A’) are all outcomes that are ______ in A. (A prime).

Probability of the Complement of an Event:
o
P( A ')  ______________
Ex. 4 A card is randomly selected from a standard deck of 52 cards. Find the
probability of the given event.
A) The card is not a king
B) The card is not an ace or a jack
Ex. 5 When 2 six-sided dice are tossed, there are 36 possible outcomes. Find the
probability of the given event (look on page 726 for the possible outcomes).
A) The sum is not 5
B) The sum is greater than or equal to 3
12.5 Probability of Independent and Dependent Events

Independent: when the first event has _________________ to do with the second
event. (ie: rolling a die twice).

Probability of Independent Events:
o
 and 
P A
B   P  A P  B 
 


Dependent: one event _______________ the occurrence of the other.

Conditional Probability: the probability that b will occur _____________ that A
has occurred. “B given A”  P  B A 

Probability of Dependent Events:
o
P  A  B   P  A P  B A

Where P  B A  
P  A  B
P  A
Ex. 1 A game machine claims that 1 in every 15 people win. What is the probability
that you win twice in a row?
Ex. 2 In a survey 9 out of 11 men and 4 out of 7 women said they were satisfied with a
product.
A) If the next 3 customers are 2 women and a man, what is the probability that they
will all be satisfied?
B) If 4 men are the next customers, what is the probability that at least one of them is
not satisfied?
Ex. 3 You randomly select 2 cards from a standard 52-card deck. Find the probability
that the first card is a diamond and the second card is red if
A) You replace the 1st card before selecting the 2nd card.
B) You don’t replace the first card.
Ex. 4
Age
5–7
8 – 10
11 – 13
Total
Attended Camp
45
94
81
No Camp
117
62
79
Total
A) Find the probability that a listed student attended camp
B) Find the probablity that a child in the 8 – 10 age group from the town did not
attend camp.
Ex. 5 Three children have a choice of 12 summer camps that they can attend. If they
randomly choose which camp to attend, what is the probability that they attend all
different camps?
Ex. 6 In one town 95 % of the students graduate from high school. Suppose a study
showed that at age 25, 81 % of the high school graduates held full-time jobs, while only
63 % of those who did not graduate held full-time jobs. What is the probability that a
randomly selected student will have a full-time job at 25?
Ex. 7 A survey showed that 47 % of students worked during the summer. Of those who
worked, 62 % watched 2 hours or more of TV per day. Those who didn’t work, 79 %
watched 2 or more hours of TV. What is the probability that you choose a random
student who watched fewer than 2 hours of TV per day.
Ex. 8 One in 10,000 cars has a defect. How many of these cars can a car dealer sell
before the probability of selling at least one with a defect is 20%?
12.6 Binomial Distributions

Binomial Experiment:
o There are a _______________ number, n, of independent trials.
o Each trial has only ___ possible outcomes…success and failure
o The probability of success is the ________ for each trial. This probability
is denoted by p. The probability of failure is given by 1 – p.

Finding a Binomial Probability:
o
P  k successes   n C k p k (1 p)n  k
Ex. 1
A) At a college, 53 % of students receive financial aid. In a random group of 5
students, what is the probablity that exactly 2 of them receive financial aid?
B) Draw a histogram of the binomial distribution for the class of students. Then find
the probability that fewer than 3 students in the class receive financial aid.


Symmetric: when the histograms left side __________________ the right side.
Skewed: ________ symmetric.
Ex. 2 Kobe Bryant’s free throw percentage is 0.856. What is the probability he will
miss 4 free throws in a game where he shoots 12 free throws.

Hypothesis Testing:
o State the hypothesis you are testing.
o Collect data from a random sample of the population and compute the
statistical measure of the sample.
o Assume the hypothesis is true and calculate the resulting probability of
obtaining the sample statistical measure.

If the probability is small (  10 %
hypothesis.
or 0.1 ) – reject the
Ex. 3 A diet company claims that 78 % of people who use their product lose weight. To
test this hypothesis, you randomly choose 9 people and ask them to do the diet for 3
months. Of the 9 people, only 4 lost weight. Should you reject this claim?